# Full text of "Analytical Mechanics"

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```308                      ANALYTICAL MECHANICS
path with a period equal to  .   The axes of the path coin-
co
cide with the coordinate axes.
Case IV.   When d = ± - and 6 = a; the path becomes a cir-
z>
cle, and the motion uniform circular motion with a period
1   4.      27T
equal to -
CO
PROBLEMS.
Find the resultant motion due to the superposition of the motions defined by the following equations:
(1)   x  a cos o}t and y = a sin at.
(2)   x  a cos (t + tQ)  and y = a sin ~~ t.
(3)   re = a sin coi and T/ = a cos  (£ + tQ).
(4)   a; = a sin (art + 5)  and y = a cos (art  5).
(5)   £ = a cos art and ?/ = 6 sin art.
(6)   a; = a sin (art  5) and y = 6 cos (art + 5).
(7)   x = a sin (art  ~) and y = 6 sin (ort + -)
\        */                          \        ^/
(8)   £ = acos(coi + ) and y = 6sin (coi  )-
\       «V                              \       «V
(9)   ^ = a cos (ut  j) and y = 6 cosf
(10)  .-r = a cos (art + 50 and ?/ = 6 sin (co^ + 5o).
241. Physical Pendulum.  Any rigid body which is free to oscillate under the action of its own weight is called a physical or a compound pendulum. Let A, Fig. 137, be a rigid body which is free to oscillate about a horizontal axis through the point 0 and perpendicular to the plane of the paper. Further let c denote the position of the center of mass and D its distance from the axis. Then the torque equation gives
/- =  mgDsmd,                         (X)
at
where m is the mass of the body and 0 the angular displacement from the position of equilibrium.```